Algebraic Identities Worksheet for Maths Olympiad

Maths Olympiad topic 30: Identities, polynomials and quadratic roots · Set 222 · 40 questions

How to solve it (Use the identity, skip the guessing): (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²; so a² + b² = (a + b)² − 2ab and (a − b)² = (a + b)² − 4ab. For x² − Sx + P = 0: the sum of the roots is S and the product is P. To factorise x² + Sx + P, find two numbers that multiply to P and add up to S.

1. 1.If a + b = 6 and ab = 8, what is a² + b²? (a) 36 (b) 28 (c) 52 (d) 20
2. 2.What is the sum of the roots of x² − 11x + 18 = 0? (a) 10 (b) 11 (c) 18 (d) 12
3. 3.Which is the correct factorisation of x² + 14x + 48? (a) (x − 6)(x − 8) (b) (x + 1)(x + 13) (c) (x + 1)(x + 48) (d) (x + 6)(x + 8)
4. 4.What is the sum of the roots of x² − 17x + 72 = 0? (a) 18 (b) 16 (c) 72 (d) 17
5. 5.Which is the correct factorisation of x² + 17x + 72? (a) (x + 8)(x + 9) (b) (x + 1)(x + 16) (c) (x + 1)(x + 72) (d) (x − 8)(x − 9)
6. 6.Using (a + b)² = a² + 2ab + b², what is 104²? (a) 9216 (b) 10816 (c) 10016 (d) 10826
7. 7.If a + b = 10 and ab = 16, what is (a − b)²? (a) 68 (b) 36 (c) 6 (d) 100
8. 8.What is the product of the roots of x² − 8x + 12 = 0? (a) 11 (b) 12 (c) 8 (d) 13
9. 9.If a + b = 18 and ab = 80, what is (a − b)²? (a) 164 (b) 4 (c) 2 (d) 324
10. 10.Which is the correct factorisation of x² + 6x + 8? (a) (x + 2)(x + 4) (b) (x + 1)(x + 5) (c) (x + 1)(x + 8) (d) (x − 2)(x − 4)
11. 11.If a + b = 19 and ab = 90, what is (a − b)²? (a) 1 (b) 361 (c) 181 (d) 271
12. 12.Using (a − b)² = a² − 2ab + b², what is 89²? (a) 8281 (b) 8099 (c) 7921 (d) 7931
13. 13.Using (a − b)² = a² − 2ab + b², what is 77²? (a) 5939 (b) 6391 (c) 5929 (d) 6889
14. 14.If a + b = 17 and ab = 60, what is (a − b)²? (a) 289 (b) 169 (c) 49 (d) 7
15. 15.If a + b = 17 and ab = 70, what is a² + b²? (a) 289 (b) 429 (c) 219 (d) 149
16. 16.If a + b = 9 and ab = 14, what is (a − b)²? (a) 81 (b) 53 (c) 5 (d) 25
17. 17.If a + b = 12 and ab = 20, what is (a − b)²? (a) 8 (b) 64 (c) 144 (d) 104
18. 18.Which is the correct factorisation of x² + 13x + 42? (a) (x + 1)(x + 42) (b) (x + 1)(x + 12) (c) (x − 6)(x − 7) (d) (x + 6)(x + 7)
19. 19.Using (a + b)² = a² + 2ab + b², what is 22²? (a) 484 (b) 404 (c) 324 (d) 494
20. 20.Which is the correct factorisation of x² + 16x + 63? (a) (x + 7)(x + 9) (b) (x + 1)(x + 63) (c) (x + 1)(x + 15) (d) (x − 7)(x − 9)
21. 21.What is the product of the roots of x² − 16x + 63 = 0? (a) 16 (b) 64 (c) 62 (d) 63
22. 22.If a + b = 18 and ab = 72, what is a² + b²? (a) 324 (b) 180 (c) 468 (d) 252
23. 23.Which is the correct factorisation of x² + 9x + 18? (a) (x + 3)(x + 6) (b) (x + 1)(x + 8) (c) (x − 3)(x − 6) (d) (x + 1)(x + 18)
24. 24.What is the sum of the roots of x² − 12x + 27 = 0? (a) 12 (b) 13 (c) 27 (d) 11
25. 25.Using (a + b)² = a² + 2ab + b², what is 53²? (a) 2819 (b) 2809 (c) 2509 (d) 2209
26. 26.Using (a − b)² = a² − 2ab + b², what is 95²? (a) 9025 (b) 9975 (c) 9035 (d) 11025
27. 27.Using (a + b)² = a² + 2ab + b², what is 105²? (a) 11025 (b) 11035 (c) 10025 (d) 9025
28. 28.Using (a − b)² = a² − 2ab + b², what is 29²? (a) 899 (b) 961 (c) 851 (d) 841
29. 29.If a + b = 6 and ab = 5, what is a² + b²? (a) 26 (b) 36 (c) 31 (d) 46
30. 30.Which is the correct factorisation of x² + 5x + 6? (a) (x − 2)(x − 3) (b) (x + 1)(x + 4) (c) (x + 1)(x + 6) (d) (x + 2)(x + 3)
31. 31.Using (a − b)² = a² − 2ab + b², what is 47²? (a) 2219 (b) 2809 (c) 2209 (d) 2491
32. 32.If a + b = 13 and ab = 30, what is (a − b)²? (a) 169 (b) 49 (c) 7 (d) 109
33. 33.Which is the correct factorisation of x² + 8x + 15? (a) (x + 1)(x + 15) (b) (x + 1)(x + 7) (c) (x − 3)(x − 5) (d) (x + 3)(x + 5)
34. 34.If a + b = 14 and ab = 24, what is a² + b²? (a) 172 (b) 244 (c) 148 (d) 196
35. 35.If a + b = 17 and ab = 66, what is a² + b²? (a) 289 (b) 421 (c) 157 (d) 223
36. 36.If a + b = 21 and ab = 108, what is a² + b²? (a) 225 (b) 657 (c) 441 (d) 333
37. 37.What is the product of the roots of x² − 6x + 8 = 0? (a) 8 (b) 6 (c) 9 (d) 7
38. 38.If a + b = 15 and ab = 36, what is (a − b)²? (a) 81 (b) 225 (c) 153 (d) 9
39. 39.If a + b = 17 and ab = 72, what is a² + b²? (a) 217 (b) 289 (c) 433 (d) 145
40. 40.What are the roots of x² − 5x + 6 = 0? (a) 2 and 4 (b) 1 and 4 (c) 2 and 3 (d) 1 and 6

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